Automated theory formation involves, amongst other things,
the production of examples, concepts and statements relating the concepts.
The HR program has been developed to form theories in mathematical domains,
by calculating examples, inventing concepts, making conjectures, and settling
conjectures using the Otter theorem prover and MACE model generator. In
addition to providing a plausible model for automated theory formation in
pure mathematics, HR has been applied to other problems in Artificial Intelligence.
We discuss HR's application to inducing definitions from examples, scientific
discovery, problem solving and puzzle generation. For each problem, we look
at how a theory formation approach can be applied and mention some initial
results from the application of HR. Our aim is not to describe the applications
in great detail, but rather to provide an overview of how HR is used for
these problems. This will facilitate a comparison of the problems and discussion
of the effectiveness of theory formation for these tasks. Our second aim
is to compare HR with the Progol machine learning program. We do this first
by looking at the concept formation these programs perform. Also, by suggesting
how Progol could be used for the applications mentioned above, we compare
the programs in terms of how they can be applied.